Second lecture: Maths tools (skimmed), ‘empirical microeconomics’, introducing preferences

Mainly for home-study: Maths and empirical tools

Partial coverage for now, some revision as we apply these

Goals of this material:

(Re)-aquaint you with some of maths tools we will use
- without scaring you

Give a flavour of what empirical microeconomics is
- and a sense of some of the key issues in empirical work

Maths are here to help you

Math tools you must know or learn (see handout)

(Univariate) functions, linear/nonlinear functions; the slope of a function (arc vs. point slope); concave/convex functions
Derivative of a function: a function that tells you the slope at each point
Minima, maxima
Functions of two or more variables, contour lines
(Simple) simultaneous equations

Slides, resources to help you, plus supplementary videos; www.khanacademy.org/math/

Lecture may skip to Mini-lecture: Empirical microeconomics/econometrics here

Univariate function

(Univariate) Function: A ‘map’ from one or more variables $x$ to an outcome $y=f(x)$

for each value of $x$ the function tells you a single value of $f(x)$; typically we assign $y=f(x)$

Linear and nonlinear functions

Linear function: A function of the form $y=a+bX$; e.g., $y = f(x) = -10 + 3x$

Plotted as a straight line; intercept $a$, constant slope $b$

Slopes

Slope of $y = f(x)$

: The change in y for a given change in x. ‘Rise over run’ $(\Delta y / \Delta x)$.

Arc slopes vs point slopes

Nonlinear (univariate) function : A function $f(x)$ of a form other than $f(x) = y=a+bX$;

E.g., a quadratic function $y = f(x) = a + bx + cx^2$
- E,g, $y = f(x) = 10 - 2x + 3x^2$

Or a logarithmic $y=ln(x)$ or exponential function $y = exp(x)$

For linear functions the slope is the same at any point. For nonlinear functions it may differ at each point.

Linear and quadratic

Instantaneous rate of change (instantaneous slope) Aka ‘point slope’: The slope of the line tangent to the curve at a single point

Convex function: Slope everywhere increasing, unique minimum where slope $=0$ Concave function: Slope everywhere decreasing, unique maximum where slope $=0$

The Derivative of a function

Derivative of a function: A derivative of a function $f(x)$ is another function called $f'(x)$.; $f'(x)$ tells us the (point) slope of the function $f(x)$ at any point $x$.

For example, the derivative of the function $f(x) = 2x + 3$ is $f'(x) = 2$
- For this linear function the slope is a constant, 2

E.g., the derivative of the quadratic function $f(x) = x^2 -4x + 1$ is $f'(x) = 2x - 4$

Plotting a function and it’s derivative

Derivative of $f(x) = x^2 -4x + 1$ is $f'(x) = 2x - 4$
- E.g., slope at $x=1$ is $f'(x;x=1) = 2\times1 - 4 = -2$
- The slope is zero where $f'(x)=2x-4=0$, or where $x=2$
- Is $x=2$ at a min, a max, or neither? How do we know?

Minimum, maximum, or neither?

$f'(x)$ is a function that tells us the slope of $f(x)$, or how $f(x)$ changes in $x$ at any point $x$

In turn, the derivative of $f'(x)$ is called $f''(x)$.

Tells us how the slope changes as $x$ increases

Oversimplifying:

slope always increasing $\rightarrow$ $f''(x)>0$ everywhere $\rightarrow$ convex (u-shaped) function $\rightarrow$ single where $f'(x)=0$
slope always decreasing $\rightarrow$ $f''(x)<0$ everywhere $\rightarrow$ concave (inverse-u) fncn $\rightarrow$ single where $f'(x)=0$

Functions of two or more variables (multivariate functions)

Utility, profit, cost, production, returns, etc.

May depend on multiple variables/inputs
Need to illustrate tradeoffs between these

\[y=f(x,z)\]

$y$ may increase and/or decrease in $x$ and in $z$,
The rate of increase of y in $x$ may depend on the values of $x$ and $z$
- Similar for the rate of increase of y in z

E.g., \[y=\sqrt(xz) = x^{1/2}z^{1/2}, x \geq 0, z \geq 0\]

Projecting a function up from X,Y space into the Z axis:

Contour lines

We will come back to this later!

Contour lines: Level sets - depict combinations of variables that hold the function constant at a particular value; f(x,z) = A for some value $A$

Level sets: E.g., indifference curves, isoquants and isocost curves.

Contour lines on a map

Consider a production function:

\[Y = f(K,L) = \sqrt(KL)\]

Setting this equal to 1 we can map out ‘all combinations of K and L that produce output $Y=1$’:

\[ Y = \sqrt(KL) = 1 \rightarrow KL = 1 \]

\[ \rightarrow K = 1/L \]

Setting this at Y = 2

\[ Y = \sqrt(KL) = 2 \rightarrow KL = 4 \] \[ \rightarrow K = 4/L \]

Simultaneous equations

E.g.,

\[ X + Y = 3 \] \[ X - Y = 1 \]

Holds only where $X=2, Y=1$, the ‘unique solution’

Meaningless to ask ‘how does a change in X affect Y?’ in the above context.

Mini-lecture: Empirical microeconomics/econometrics

Empirical research: Uses evidence from the real world, i.e., observation, to answer questions

Econometrics: The ‘science’ of using data to answer economic questions. Uses statistical tools and often economic theory

Micro-data: Data where the unit of observation is an individual, household, firm, etc.

Empirical(ish) example

Trying to estimate demand curve, hypothesize linear function \[Q_d = a-bp\]

Suppose we know price is shifting because of costs, shifts in supply curve, or the firm is experimenting. Observe price & quantity data for a period where ceteris paribus is reasonable.

Fit ‘best’ line (minimise error) through these points

Estimate demand slope & intercept, use to make inferences
Never fits exactly. why not?

Ceteris paribus

All [most] economic theories employ the assumption that ‘other things are held constant.’

Above, demand may differ between weeks/stores, weather changes, etc.

the points may lie on several different demand curves, and attempting to force them into a single curve would be a mistake.

$\rightarrow$ Carefully ‘control’ for other observable factors (a partial solution)

But what if we just observe a dataset of ‘price and quantity’ in a large market?

Can we estimate a demand curve?
A supply curve? Both?

Application 1A.3: … Changing world oil prices (time-permitting)

Empirical work has estimated:

\[ Q = 85 - 0.4P \: (D) \] \[ Q = 55 + 0.6P \: (S) \]

Solving: $85 - 0.4P=55+0.6P \rightarrow P = 30, Q = 73$

Approximates 2000-2002 price

Why did price rise to US$130 in 2008 and fall to $50 by March ’09?

China & India’s economies grew $\rightarrow$ growth in the world economy by 3-4% per year
- (Various calculations) $\rightarrow$ Demand shifts out from $Q_D = 85 - 0.4P$ to:

\[Q_D = 112 - 0.4P\] \[Q_S = 55 + 0.6P\]

$\rightarrow$ solves to $P=57, Q=87$

Overall price inflation, US$ devaluation $\rightarrow$ gets us to about $94. So why the $130 price?

First problem set: coverage

See Handout/web-book

Representative answers for each problem set given about 1-week after posting
Five support classes (tutorials), cover parts of these

E.g., …

Plotting supply and demand “for orange juice”, solving, for equilibrium price, excess demand/supply at non-equilibrium prices

Example of some (tricky) MCQs from previous exams

E.g., “True or false: It is valid to plot observed prices and quantities traded in a market and fit a line through them to estimate a market demand curve.”

Discussion questions: practice writing concise essays and bullet points

Preamble to Utility and Choice

(NS - Chapter 2)

Motivation

Consider a decision you recently made?

Define this decision clearly.
How do you think you decided among these options?

2 minutes: discuss with your neighbour

Suppose I asked you

‘State a rule that governs (determines/characterizes) how people do make decisions’…

I want this rule to be…

Informative (it rules out at least some sets of choices)
Predictive (people rarely if ever violate this rule)